For my state, for 2008 property tax, we pay in May of 2009. So if I bought a house in Oct. 2008, the former owner will pay me the tax from Jan to oct. of 2008. This money would be used for my closing cost. In May, 2009, I would get a bill from the city about the 2008 tax, I would pay all tax of the house for 2008. This is the time for me to pay 2008 tax. So I can only claim that I pay property tax in 2009, even it is the tax for 2008. ( Keep in mind that your tax is only the part of 2008, even your tax bill is for the full year. The former owner paid the rest.)
I hope this is clear for you now.